The question is incomplete! Complete question along with answer and step by step explanation is provided below.
Question:
A class of fourth graders takes a diagnostic reading test, and the scores are reported by reading grade level. The 5-number summaries for the 14 boys and 11 girls are shown:
Boys: 2.0 3.9 4.3 4.9 6.0
Girls: 2.8 3.8 4.5 5.2 5.9
a) Which group had the highest score?
b) Which group had the greater range?
c) Which group had the greater interquartile range?
d) Which group’s scores appear to be more skewed? Explain.
A. The boys' scores are more skewed, because the quartiles are the same distance from the mean.
B. The girls' scores are more skewed, because the quartiles are the same distance from the median. (In the distribution of boys' scores, the quartiles are not the same distance from the median.)
C. The girls' scores are more skewed, because the quartiles are not the same distance from the median. (In the distribution of boys' scores, the quartiles are the same distance from the median.)
D. The boys' scores are more skewed, because the quartiles are not the same distance from the median. (In the distribution of girls' scores, the quartiles are the same distance from the median.)
a) Highest score = Boys = 6
b) Greatest Range = Boys = 4
c) Greatest Interquartile Range = Girls = 1.4
d) D. The boys' scores are more skewed, because the quartiles are not the same distance from the median. (In the distribution of girls' scores, the quartiles are the same distance from the median.)
Step-by-step explanation:
Let us first understand what is a 5-number summary!
A 5-number summary refers to a box plot which basically shows 5 statistical characteristics of a data set and they are
1. minimum value
2. Lower quartile
3. Median value
4. Upper quartile
5. Maximum value
Boy's Group:
Minimum value = 2
Lower quartile = 3.9
Median value = 4.3
Upper quartile = 4.9
Maximum value = 6
Girl's Group:
Minimum value = 2.8
Lower quartile = 3.8
Median value = 4.5
Upper quartile = 5.2
Maximum value = 5.9
Now we can proceed to answer each question!
a) Which group had the highest score?
The highest score is given by the maximum value therefore, boys group has a higher score (6) as compared to girls group (5.9).
Highest score = Boys = 6
b) Which group had the greater range?
The range is given by maximum value - minimum value.
Range of Boy's Scores:
Range = maximum value - minimum value
Range = 6 - 2
Range = 4
Range of Girl's Scores:
Range = maximum value - minimum value
Range = 5.9 - 2.8
Range = 3
Therefore, boys group had the greater range of (4) as compared to the girls group (3).
Greatest Range = Boys = 4
c) Which group had the greater interquartile range?
The interquartile range is given by upper quartile - lower quartile.
Interquartile Range of Boy's Scores:
Interquartile Range = upper quartile - lower quartile
Interquartile Range = 4.9 - 3.9
Interquartile Range = 1
Interquartile Range of Girl's Score:
Interquartile Range = upper quartile - lower quartile
Interquartile Range = 5.2 - 3.8
Interquartile Range = 1.4
Therefore, girls group had the greater interquartile range of (1.4) as compared to the boys group (1).
Greatest Interquartile Range = Girls = 1.4
d) Which group’s scores appear to be more skewed? Explain.
A data set is said to be skewed when the distance between interquartiles and the median is not equal.
Boy's Scores:
The distance between upper quartile and median: 4.9 - 4.3 = 0.6
The distance between median and lower quartile: 4.3 - 3.9 = 0.4
Girl's Scores:
The distance between upper quartile and median: 5.2 - 4.5 = 0.7
The distance between median and lower quartile: 4.5 - 3.8 = 0.7
As you can notice, boys group is more skewed than the girls group since the distance between the median and the interquartiles is not equal in the boys case.
On the other hand, we can say that the girls group is symmetric as compared to the boys group since the distance between interquartiles and the median is equal.
Therefore, the correct option is D.
D. The boys' scores are more skewed, because the quartiles are not the same distance from the median. (In the distribution of girls' scores, the quartiles are the same distance from the median.)
